scispace - formally typeset

Hadamard code

About: Hadamard code is a(n) research topic. Over the lifetime, 1049 publication(s) have been published within this topic receiving 12130 citation(s). The topic is also known as: Walsh code & Walsh–Hadamard code. more


MonographDOI: 10.1515/9781400842902
Abstract: In Hadamard Matrices and Their Applications, K. J. Horadam provides the first unified account of cocyclic Hadamard matrices and their applications in signal and data processing. This original work is based on the development of an algebraic link between Hadamard matrices and the cohomology of finite groups that was discovered fifteen years ago. The book translates physical applications into terms a pure mathematician will appreciate, and theoretical structures into ones an applied mathematician, computer scientist, or communications engineer can adapt and use. The first half of the book explains the state of our knowledge of Hadamard matrices and two important generalizations: matrices with group entries and multidimensional Hadamard arrays. It focuses on their applications in engineering and computer science, as signal transforms, spreading sequences, error-correcting codes, and cryptographic primitives. The book's second half presents the new results in cocyclic Hadamard matrices and their applications. Full expression of this theory has been realized only recently, in the Five-fold Constellation. This identifies cocyclic generalized Hadamard matrices with particular "stars" in four other areas of mathematics and engineering: group cohomology, incidence structures, combinatorics, and signal correlation. Pointing the way to possible new developments in a field ripe for further research, this book formulates and discusses ninety open questions. more

Topics: Complex Hadamard matrix (71%), Hadamard matrix (71%), Hadamard three-lines theorem (67%) more

492 Citations

Open access
Jennifer Seberry1, Mieko Yamada1Institutions (1)
01 Jan 1992-
Abstract: One hundred years ago, in 1893, Jacques Hadamard [31] found square matrices of orders 12 and 20, with entries ±1, which had all their rows (and columns) pairwise orthogonal. These matrices, X = (Xij), satisfied the equality of the following inequality, |detX|2 ≤ ∏ ∑ |xij|2, and so had maximal determinant among matrices with entries ±1. Hadamard actually asked the question of finding the maximal determinant of matrices with entries on the unit disc, but his name has become associated with the question concerning real matrices. Disciplines Physical Sciences and Mathematics Publication Details Jennifer Seberry and Mieko Yamada, Hadamard matrices, Sequences, and Block Designs, Contemporary Design Theory – A Collection of Surveys, (D. J. Stinson and J. Dinitz, Eds.)), John Wiley and Sons, (1992), 431-560. This journal article is available at Research Online: 11 "-. Hadamard Matrices, Sequences, and Block Designs Jennifer Seberry and Mieko Yamada 1 IN1RODUCTION 2 HADAMARD MATRICES 3 THE SmONGEST HADAMARD CONSmUCTION THEOREMS 4 ORTIIOGONAL DESIGNS AND AsYMPTOTIC EXISTENCE 5 SEQUENCES 6 AMICABLE HADAMARD MAmICES AND AOD 7 CoNSmUCTIONS FOR SKEW HADAMARD MAmICES 8 M -SmucTUREs 9 WILLIAMSON AND WILUAMSON-TYPE MAmICES 10 SBIBD AND THE EXCESS OF HADAMARD MATRICES 11 CoMPLEX HADAMARD MATRICES APPENDIX REFERENCES more

Topics: Hadamard transform (62%), Hadamard code (61%)

303 Citations

Open accessJournal ArticleDOI: 10.1109/LPT.2004.833859
Abstract: A new code structure for spectral-amplitude-coding optical code-division multiple-access system based on double-weight (DW) code families is proposed. The DW code has a fixed weight of two. By using a mapping technique, codes that have a larger number of weights can be developed. Modified double-weight (MDW) code is a DW code family variation that has variable weights of greater than two. The newly proposed code possesses ideal cross-correlation properties and exists for every natural number n. Based on theoretical analysis and simulation, MDW code is shown here to provide a much better performance compared to Hadamard and modified frequency-hopping codes. more

Topics: Constant-weight code (72%), Systematic code (69%), Polynomial code (69%) more

274 Citations

Open accessJournal ArticleDOI: 10.1007/S11080-006-8220-2
Abstract: Complex Hadamard matrices, consisting of unimodular entries with arbitrary phases, play an important role in the theory of quantum information. We review basic properties of complex Hadamard matrices and present a catalogue of inequivalent cases known for the dimensions N = 2,..., 16. In particular, we explicitly write down some families of complex Hadamard matrices for N = 12,14 and 16, which we could not find in the existing literature. more

Topics: Complex Hadamard matrix (79%), Hadamard matrix (73%), Hadamard three-lines theorem (71%) more

273 Citations

Open accessJournal ArticleDOI: 10.1214/AOS/1176344370
Abstract: An $n \times n$ matrix $H$ with all its entries $+1$ and $-1$ is Hadamard if $HH' = nI$. It is well known that $n$ must be 1, 2 or a multiple of 4 for such a matrix to exist, but is not known whether Hadamard matrices exist for every $n$ which is a multiple of 4. The smallest order for which a Hadamard matrix has not been constructed is (as of 1977) 268. Research in the area of Hadamard matrices and their applications has steadily and rapidly grown, especially during the last three decades. These matrices can be transformed to produce incomplete block designs, $t$-designs, Youden designs, orthogonal $F$-square designs, optimal saturated resolution III designs, optimal weighing designs, maximal sets of pairwise independent random variables with uniform measure, error correcting and detecting codes, Walsh functions, and other mathematical and statistical objects. In this paper we survey the existence of Hadamard matrices and many of their applications. more

Topics: Hadamard matrix (77%), Hadamard transform (67%), Hadamard code (64%) more

272 Citations

No. of papers in the topic in previous years

Top Attributes

Show by:

Topic's top 5 most impactful authors

Christos Koukouvinos

9 papers, 120 citations

Hilal A. Fadhil

5 papers, 119 citations

Hakob Sarukhanyan

5 papers, 12 citations

Ali N. Akansu

5 papers, 19 citations

Martin Bossert

4 papers, 29 citations

Network Information
Related Topics (5)
Decoding methods

65.7K papers, 900K citations

82% related
Channel capacity

16.8K papers, 503.3K citations

82% related
Linear code

16.3K papers, 448.6K citations

82% related
Low-density parity-check code

19.8K papers, 331.8K citations

82% related
Sequential decoding

8.6K papers, 204.2K citations

81% related